1 / 26

Regularity partitions and the topology of graphons

Regularity partitions and the topology of graphons. L á szl ó Lov á sz Eötvös Lor ánd University, Budapest . Joint work Bal á zs Szegedy. The Szemerédi Regularity Lemma. given ε >0, # of parts k satisfies 1 / ε  k  f ( ε ). difference at most 1. with ε k 2 exceptions.

vidar
Download Presentation

Regularity partitions and the topology of graphons

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Regularity partitions and the topology of graphons László Lovász Eötvös Loránd University, Budapest Joint workBalázs Szegedy

  2. The Szemerédi Regularity Lemma given ε>0, # of parts k satisfies 1/ ε kf(ε) difference at most 1 with εk2 exceptions for subsets X,Y of the two parts, # of edges between X and Y is pij|X||Y| ε(n/k)2 The nodes of  graph can be partitioned into a bounded number of essentially equal parts so that almost all bipartite graphs between 2 parts are essentially random (with different densities pij).

  3. The Szemerédi Regularity Lemma Original Regularity LemmaSzemerédi 1976 “Weak” Regularity LemmaFrieze-Kannan 1999 “Strong” Regularity LemmaAlon – Fisher - Krivelevich - M. Szegedy 2000 Tao 2005 L-Szegedy 2006

  4. The many facets of the Lemma Low rank matrix approximation - Frieze-Kannan Probability, information theory - Tao Approximation theory - L-Szegedy Sparse Regularity Lemma Gerke, Kohayakawa, Luczak, Rödl, Steger, Arithmetic Regularity Lemma Green, Tao Regularity Lemma and ultraproducts Elek, Szegedy Hypergraph Regularity Lemma Frankl, Gowers, Nagle, Rödl, Schacht Measure theory - Bollobás-Nikiforov Compactness - L-Szegedy Dimensionality - L-Szegedy

  5. Graphons Could be:

  6. Pixel pictures 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 G AG WG

  7. Cut distance of graphons cut norm measure preserving cut distance

  8. Regularity Lemma and cut distance P: measurable partition of [0,1], Weak Regularity Lemma: Strongest Regularity Lemma: is compact

  9. Subgraph density Probability that random map V(F)V(G) is a hom

  10. Graphons as limit objects Borgs, Chayes, L, Sós, Vesztergombi

  11. Graphons as limit objects For every convergent graph sequence(Gn) there is a graphon such that Conversely, for every graphon Wthere is a graph sequence (Gn) such that L-Szegedy Wis essentially unique (up to measure-preserving transformation). Borgs- Chayes- L

  12. Example: Randomly growing graphs A randomly grown uniform attachment graphwith 200 nodes

  13. Example: Generalized random graph Random with density 1/3 Random with density 2/3 Random with density 1/3

  14. Example: Borsuk graphon Sdwith uniform distribution W(x,y)=1(|x-y|>2-d-2) Gn: induced subgraph on n random nodes Gn W F3:  Gn has weak regularity partition with O(d) classes  W has a d-dimensional „underlying space”  Neighborhoods in Gn have VC-dimension d+1  Gn does not contain Fd+1 as an induced subgraph

  15. The topology of a graphon w t u s v Squaringtheadjacencymatrix

  16. The topology of a graphon After a lot of cleaning Complete metric spaces A = Borel sets has full support Not always compact Compact pure graphon

  17. Example: Generalized random graphs again (J,rG): discrete (J,rW)  (J,rWoW): (J,rGoG): (J,rWoW): Gromov-Wasserstein convergence

  18. Example: Borsuk graphon again Gn: induced subgraph on n random nodes G’n: randomly delete half of the edges from Gn

  19. Regularity and dimensionality SJVoronoi cells of S form a partition with  partition P={V1,...,Vk}of [0,1]  viVi with average ε-net regular partition

  20. Regularity and dimensionality Theorem. ε>0, the metric space (J,rWoW) can be partitioned into a set of measure <ε, andsets with diameter <ε.

  21. Extremal graph theory and dimensionality F: bipartitegraphwithbipartition (U,V), G: graph W: graphon F bi-induced subgraph of G: U’,V’V(G), disjoint, subgraph formed by edges between U’ and V’ is isomorphic to F F bi-induced subgraph of W:

  22. Extremal graph theory and dimensionality Theorem. F is not a bi-inducedsubgraph of W  W is 0-1 valued, (J,rW) is compact, and has finitepackingdimension. Key fact: VC-dimension of neighborhoods is bounded

  23. Extremal graph theory and dimensionality Corollary. P: hereditary bigraph property not containing all bigraphs. (J,W): pure graphon in its closure  W is 0-1 valued,(J,rW) is compact and has bounded dimension.

  24. Extremal graph theory and dimensionality Corollary. P: hereditary graph property not containing all graphs, such that W in its closure is 0-1 valued, (J,W): pure graphon in its closure  (J,rW)is compact and has bounded dimension.

  25. Example P: triangle-free

  26. Extremal graph theory, dimensionality and regularity Corollary. F is not a bi-induced subgraph of G  >0, G has a weak regularity partition with error  with at most classes.

More Related